is the Japanese bracket,
\(\alpha =(\alpha _1,\alpha _2)\in \mathbb {N}_0^2\)  is a multi-index with \(|\alpha |=\alpha _1+\alpha _2\) 
and \(\partial ^\alpha \)  is a short notation for \(\partial ^{\alpha _1}_{x_1}\partial ^{\alpha _2}_{x_2}\) .
the space \({S}_{B}\)  is closed and the algebraic operations are continuous with respect to the
Fréchet topology of the Schwartz functions. A proof of these well-known facts can be found in [1]} or in [2]}. The density of \({S}_{B}\)  in
\({L}^1_B\)  follows from the density of \({S}\big (\mathbb {R}^2\big )\) 
in \(L^1\big (\mathbb {R}^2\big )\)  [3]}. In summary one has that
